"Skipping orbits traversing trajectories; quantum ballistic transport in microstructures"

Superlattices and Microstructures, Vol. 5, No. 7, 1989

127

SKIPPING ORBITS, TRAVERSING TRAJECTORIES, AND

QUANTUM BALLISTIC TRANSPORT IN MICROSTRUCTURES

C. W.J. Beenakker and H. van Honten"

Philips Research Laboratories, 5600 JA Eindhoven, The Netherlands

B..f. van Wees

Department of Applied Physics, Delft University of Technology

2600 G A Delft, The Netherlands

(Received 8 August 1988)

Three topics of current interest in thc study of quantum ballistic transport in a

two-dimensionai clcctron gas arc discussed, with an cmphasis on correspondences

bctwcen classical trajectories and quantum statcs in thc various expcrimental

ge-ometrics. Wc consider the quantized conductancc of point contacts, the quenching

of thc Hall effect in narrow channels, and coherent elcctron focusing in a

double-point contact geometry.

1. Introduclion

Quantum ballistic transporl in a two-dimensionnl

clcclron gas (2DHG) is a fascinating ncw field of

rc-scarch, cnablcd by advanccs in molecular beam

cpitaxy and microfabrication techniques. On thc onc

hand, GaAs-AlGaAs hctciostructures can bc grovvn

which havc vcry littlc impurity scattering in thc

2DEG, so that large mcan frcc paths, on the order of

10 fim, arc realizcd. Motion of thc clcctrons on this

Icngth scalc procccds along ballistic trajectories

in-volving repcatcd collisions wilh thc boundary. On thc

other hand, it has bccomc possiblc to fabricatc

microstructurcs with minimal dimcnsions comparablc

to thc De Broglie wavc length λ/.· ~ 50 nm of the

currcnt-carrying clcctrons al thc Perm i Icvcl. On this

Icngth scalc thc intcrfercncc of clectrons moving o n

diffcrcnt trajcctorics leads to intcrcsting quantum

phcnomcna. Threc rcccntly discovcrcd cxamples are

rcviewed in this article.

Our discussion is in tctms of two alternative ways

of trcating quantum ballistic transport through a

2DEG channcl: Eithcr in tcrms of intcrfcring

trajec-tories (äs in a Feynman path integral), or in tcrms of

a discretc set of quantum stalcs or l D subbands.

These two äquivalent ways of dcscriplion arc

analo-gous to the ray versus modc dcscription of

propa-gation t h r o u g h an oplical fibcr or wavc guide. We

have found this analogy with optics fruitful, both to

understand the expcrimcnts and to inspire ncw oncs.

In the semi-classical approximation o n l y interferenccs

of classical trajectories arc retained. This is equivalcnt

a. Prescnt addrcss: Philips Laboratories, Briarcliff

Manor, NY 10510, USA.

to solving l he Schrödingcr cquation in thc W K B

ap-p r o x i m a t i o n . Thc q u a n t u m statcs are thcn simap-ply

givcn by thc Bohr-Sommcrfcld quantization rulc. The

characlcr of thc q u a n t u m states can be conlinuously

changccl by applying an cxtcrnal magnctic ficld,

ori-cntecl pcrpcndicular to the 2DEG. Wc will discuss a

"phase diagram", in which thc q u a n t u m states (and

Lhc concsponding trajectories) arc classificd according

to tlicir lateral cxtcnsion in cdgc statcs (skipping

or-bits), travcrsing statcs, and Landau levcls (cyclotron

orbits). Wc bclicvc t h a t many csscntiul fcaturcs of

q u a n t u m ballistic transport can be undrrstood on the

basis of I h i s simple classification.

Thc thrcc cxamples considcred all involve

trans-port through microstructurcs dcfined in the 2DEG of

a GaAs-AlGaAs hctcrostructure. We first consider

point contacts, in scction 2. The icsidual resistance

in thc ballistic transport regime of a short and narrow

channcl connccting two broad regions (a point

con-tact) is due to clcctrons which are rcflected at the

channel cntrancc. In mctals, this resistance is known

äs thc Sharvin contact resistance

1

, and can be

de-scribcd classically sincc thcrc λ/7 ~ 0.5 nm is much

smaller than achicvable point contact widths. In the

2DEG, howcvcr, λρ is a hundrcd times äs large, a

length scale which is within rcach of lithographical

techniques. This has cnabled our group

2

, and

inde-pcndently a group from thc Cavcndish laboratory

3

, to

fabricate a q u a n t u m point contact (QPC) of variable

width comparablc to λρ. Wc discuss the origin of the

conductancc quantization in a QPC in terms of the

analogy with an clcctron wave guide

4

. In section 3

wc consider thc correspondences bctwcen quantum

statcs and classical trajectories in a narrow 2DEG

channcl in a wcak magnctic ficld, and discuss a

pos-sible thcorclical cxplanation

5

for the quenching of the

128

Superlattices and Microstructures, Vol. 5, N o 1, 1989

-2 -l θ -l 6 -l 4 -l 2 -l

GATE VOLTAGE (V)

Fig.l. Point contact conductancc (coirected foi a

series lead rcsistance) äs a function of gatc voltage

for scveral magnctic ficld values, illustrating Ihe

transition from zero-ficld quantization to quanlum

Hall effect. The curves havc bcen offset for clarity.

The inset shows the devicc geometry, with the

dc-pletion rcgions dcfining the point contact inclicated

schematically. (Fig. takcn from Ref. 2.)

Hall effect discovered cxpeiimentally by Roukcs et

αϊ.6. Finally, in section 4, we consider coherent

electron focusing

7

(CEF) in a geometry involving two

adjacent point contacts on a single boundary of the

2DEG. This experimenl allows one to study the

in-terferencc of skipping orbits along the 2DEG

boundary

8

. From a diffcrcnt point of vicw, CEF is a

typical cxamplc of a non-local voltage measurcment,

which providcs a demonstration of the reciprocity

re-lation for non-local phase-cohcrcnt transport dciived

by Büttiker

9

.

2. Quantum point contacts

The QPC is a narrow and short channel of

vari-able width W ~ λ r ~ 50 nm, dcfmcd in the 2DEG

by applying a negative voltage on a split gate on top

of the hcterostructure (see Fig. l , inset). The channel

Icngth L> IVis much smallcr than the mcan fice path

/ ~ ΙΟμιη. As discovercd icccntly

2

·

3

, the contact

conductancc G

c

of a QPC is approximatcly quantizcd

in units of 2e

2

//?, without α mognetic ßeid. If a

mag-netic ficld is applicd pcipcndicular to the 2DEG, a

continuous transition to the quantum Hall effect is

observed (Fig. 1). Additional plateaus at odd

multi-ples of e

2

jh are resolved abovc ficlds of about 2 T, äs

the magnctic ficld rcmovcs the spin dcgencracy.

[These additional plateaus arc also resolved in paiallcl

ficlds

3

, but much higher fields cxcceding 10 T are

ic-quired; This may be duc to thc anisotropic

cnhancc-ment of the Lande g— factor in quasi l D channcls

found by Smith et a/.

1

".] In Ref. 2 wc gave a

scmi-classical cxplanation" of the zcro-ficld quantization,

based on thc assumption of quantizcd transversc

mo-mentum in thc QPC, and discussed the fundamental

rclation betwcen contact tesistanccs and L a n d a u c i ' s

f o r m u l a '

2

which was pointcd out by Imry'

3

.

Table 1.

The electron wave guide

ray

modc

modc indcx

wavc numbcr k

frequcncy ω

dispcrsion law <o(k)

group velocity dwjdk

·«*· tiajectory

<=> subband

<» q u a n t u m numbcr n •»canonical momcnlum hk <*· cncrgy ε = hco

·» band structuic e

n

(k)

·»· velocity

In tcrms of the wave guide analogy (Tablc I), thc

conductancc quantization aiiscs bccausc the current /?

shared equally among an integer numbcr of exriled

modes, despitc thc fact lhat d i f f c i c n t mocles

n=\,2,...N have diffcrcnt group vclocitics

vn=dB„lhdk. Thc point is that thc group velocity

cancels with thc density of statcs p„ = (nde,„ /<*)"'

(both cvaluatcd at thc Fcrmi eneigy), so that thc

cui-rent per mode is ev

n

p„ eV = (2i'

2

//?)K — rcgardlcss of

cncrgy or modc index. The conductancc, which is thc

total current dividcd by thc appiicd voltage V, thcn

bccomes

G, =

( I )

äs observed expcrimentally

2

'

3

.

Superlattices and Microstructures, Vol. 5, No. 1, 1989

129

ciuantization is thcreforc m u c h Icss robust than thc

q u a n l u m Hall cffcct, and is not likcly to providc an

alternative resistancc Standard. The point is, äs

cm-phasized b}' Büttiker

1 5

, t h a t a largc magnctic ficld

suppresscs back-scattcring by spatially scparating

Icft-and right-moving elcctrons at oppositc cdgcs of thc

channcl. One furthcr distinction from the q u a n l u m

H a l l effcct is that, in principle, Eq. (1) is not restrictcd

to two dimensions, but also holds for a 3D wirc with

transvcrsc dimensions of ordcr λ/,· .

Thc contact conductancc Gr givcn by Eq. (1)

rc-fcrs to a two-tcrmina! mcasurcmcnt, C/,, =//ί'Δμ,

whcre thc chcmical potcntial diffcrcncc Λμ is

mcas-urcd betvvccn thc sourcc and d r a i n for thc currcnl /

(Fig. I, insct). This q u a n t i t y docs not conlain

Infor-mation on the spatial distribulion of thc voltagc drop.

Such Information can be oblaincd from thc

four-terminal resistancc RM = ρ(μ, — μκ)/1, dcfmccl in

tcrms of thc chcmical potcnlials fi; and μ

κ

mcasurcd

by two voltagc p rohes at oppositc sidcs of thc

con-striction (Fig. 2, insel). Mcasurcmcnts

1 6

of Λ

4

, havc

found a negative magnctorcsistancc which (äs shown

in Fig. 2) is well dcscribed by thc Landaucr-typc

formula"

1

/? — ' 'Mi — Γ

2p2 N /V,.

(2)

whcre Nj, = kF/^ά /2 is thc numbcr of occupicd

Landau levcls in thc broad rcgions acljacent to thc

constriction, which itsclf has N occupicd subbands.

[A similar formula has indcpcndcntly bcen obtaincd

by Büttiker

1 5

.] Thc negative magnctorcsistancc

prc-dictcd by Eq. (2) results from rcduccd back-scaltcring

at thc cntrance of thc constriction. As indicatcd

schc-matically in Fig. 2 (insct), right- and Icft-moving

clcctrons in thc broad rcgions arc spatially scparatcd

by a magnclic ficld. This rcduccs thc probability that

clcctrons approaching thc constriction arc scattcrcd

back into the broad rcgions, and Icads to a dccrcasc

of /?4, with incrcasing B , u n t i l 2/

cycl

< W. Thcn

N = Nj so that Λ

4

, = 0, äs if Ihc constriction wcrc not

therc. The two-tcrminal resistancc l/G

r

(Eq. (1)) docs

not vanish, of coursc, bul bccomcs idcntical to thc

Hall rcsistance R

f /

,

h

2e2

l

Λ', (3)

[Note that, in Ihc cxpcrimcnt shown in Fig. 2, a

rc-duced clcctron dcnsity in the constriction Icads to a

crossovcr to a positive magnctorcsistancc at higher

ficlds, in accordancc with Eq. (2).] One could still call

R/, a contact rcsistance, originatin g at thc sourcc and

drain wherc clcctrons entcr or leavc thc 2DEG with

an cffcctivc "conductivc width" of ordcr /

cyc

i. This

point of vicw is supportcd by thc obscrvations ·

3

(Fig.

I) of a continuous transition in thc two-tcrminal

rc-sistance from zcro-ficld q u a n t i z a t i o n to q u a n l u m Hall

cffcct.

Thc spccial rolc of contact rcsistanccs was not

apprcciated in this ficld u n t i l rcccntly. Thc

Landaucr-typc formula G = (2r

2

//;)/V(l —r), which implics Eq.

2000

1500 1000 500 -0.6 -0.4 -0.2 0 B(T) 0.2 0.4 0.6

Fig.2. Four-terminal magnctoresistance of a

con-striction for a scries of gate voltagcs from 0 V

(lowcst curvc) to —3 V. Solid lines are according to

Eq. (2), with the constriction width äs adjustable

parametcr. The negative magnetorcsistance is thc

rcsult of rcduccd back-scattering in a magnetic

ficld. Thc insct shows thc dcvicc gcomctry. Thc

spatial Separation by a magnctic field of left- and

right-moving electrons is indicated. (Fig. takcn

from Ref. 16.)

(1) in the absencc of back-scattcring (r — 0) , has long

bccn subject to controvcrsy — since it was not

undcr-stood whcre the residual rcsistance of a perfecl

con-ductor came from'

7

. Indced, the original

(one-dimcnsional) Landauer

1 2

formula G =

(2i>

2

//;Xl -r)/rgives l/G

1

= 0 for r = 0 . The qucstion

of contact rcsistances was settlcd by Imry

1 3

, just

be-fore acquiring an uncxpcctcd significance with the

QPC cxpcrimcnts.

3. Quenching of the Hall effect

Thc familiär cxprcssion (3) for the Hall resistance

is valid only in a broad 2DEG. Recent

mcasurcmcnts

6

' of R

fi

for ballistic transport through

narrow 2DEG channcls havc shown deviations from

Eq. (3) at sufficiently low magnetic fields. The

exper-imcnts can bc describcd in terms of two field scales.

Firstly, deviations from a linear B — dependence of

RH dcvclop below a field B^. Secondly, in the

narrowcst channcls and at low temperatures a

re-m a r k a b l c platcau of zcro Hall resistance is found

be-low a thrcshold magnctic ficld ßihres· This is the

phcnomcnon of the qucnching of thc Hall effect. In

Ref. 5 it was argued that thcse two field scales are

givcn approximatcly by

Alircs :

(4α)

130

Superlattices and Microstructures, Vol. 5, No. 1, 1989

The field 5

crU

is reached when the channel width W

is of the order of the cyclotron diameter, and the field

5

thres

when W is of the order of the transverse wave

length of magnetic edge states. Good agreement was

obtained with the experiments of Roukes et αϊ.

6

, οη a

series of etched wires of different widths. Ford et

ö/.

18

have recehtly reported significant disagreement

with Eq. (4b) in a 2DEG channel of variable width,

defmed electrostatically by a gate potential.

Uncer-tainties in the dependcnce of W and k

F

on the gate

Potential, combined with the sensitivity of Eq. (4b) to

the precise value of W (because of the iV~

3

power

law), may account for part of the disagreement.

Clearly further experiments on wcll-dcfined Systems

are necessary to settle the issue.

The arguments of Ref. 5 are based on the

diffcr-ences in lateral extension of states at the Fermi level,

when a 2DEG channel is placed in a perpendicular

magnetic field. The physics involved is conveniently

discussed in terms of a "phase diagram" (Fig. 3),

which illustrates the classical .correspondenccs of the

various quantum states. Classically, we can

distin-guish three types of trajectories in a magnetic field,

dcpending on the cnergy ε (or cyclotron radius

(2/778)'

/2

/pß) and the Separation X of the cyclolron

orbit centcr from the line je = 0 in the middle of the

channel. These are: 1. Circular cyclotron orbits,

which correspond to Landau levels; 2. Skipping

or-bits, corresponding to edge states; and 3. Traversing

trajectories, corresponding to states which interact

with both boundaries. In the (X, e) space the different

types of trajectories are separated by two parabolas

(Fig. 3). The quantum mechanical dispersion law

6„(/c) can be drawn into this classical "phase diagram",

because of the correspondence k = — X eB\h . [ This

correspondcnce exists because both k and X are

con-stants of the motion, and follows from the fact that

the canonical momentum hk along the channel equals

hk = mv

y

—eA

y

= mv

y

—eßx — — eBX,

in the Landau gauge A = (Ο,Αχ,Ο). ] We have done

this in Fig. 4 for values of B, W, and k

F

(taken from

Ref. 6) in each of the three rcgimes B > Β

α

·

Λ

,

ßthres < B< ß

C T i t

, a n d S <S

thres

.

If B > Β^-,ι (Fig. 4a) there are no states at the

Fermi level which interact with both the opposite

cdges of the channel. Consequently, the Hall

resist-ance takes its normal value (Eq. (3)) for a broad

2DEG. If S

thres

< B < ß

crit

(Fig. 4b) there are, in

addition to edge states on each of the boundaries, also

states at the Fermi level which interact with both

cdges. In this regime classical size effects lead to

devi-ations from Eq. (3). Finally, if B < 5

lhres

(Fig. 4c)

there are at the Fermi level only states which interact

with both edges. All edge states are suppresscd, since

thcir transverse wave length exceeds the channel

width. As argued in Ref. 5, this suppression of edge

states could lead to a vanishing Hall resistancc. The

argument is based on Büttiker'iT four-tcrminal

rcsist-ance formula, which relates resistrcsist-ances to

trans-mission probabilities into voltage probes. This

formula implies a vanishing Hall resistance if an

Fig.3. Energy — orbit center phase space. The two

parabolas dividc the space into four regions which

correspond to different types of classical

trajccto-ries in a magnetic field (clockwise from left:

skip-ping orbits on onc edge, traversing trajectories,

skipping orbits on the other edge, and cyclolron

orbits). The shadcd area is forbiddcn. The region

at the upper centcr contains traversing trajectories

moving in both dircctions, but only one dircction

is shown for clarity.

electron moving along the 2DEG channel has equal

probability of entering one or the othcr of two

oppo-site voltage probes. Since traversing trajectories

col-lide with equal frequency on both channel boundaries,

they do not contribute to the Hall voltage, so that

skipping orbits are a classical prcrequisite for a

non-zero R,]. The classical correspondence then suggests a

quenching of the Hall eff'ect once all cdgc states are

suppresscd. A dctisivc tost of this argument

5

would

bc a numerical calculation of the transmission

proba-bilities in a magnetic field. This has not yet been done.

Peeters

19

has shown that no quenching occurs if the

vollage probes are weakly couplcd by tunnel junctions

to a conducting channel (this case can be solvcd

ana-lytically). The experiments are, howcver, performed in

the opposite limit of strong coupling, whcre an

electron has a large probability of being diverted into

one of the voltage probes

20

. The negative rcsult of

Ref. 19 is therefore by itself not in conflict with the

experiments, nor with Ref. 5 (where coupling to the

voltage probes via ballistic motion, rather than

tunneling, is assumcd).

4. Coherent electron focusing

Skipping orbits can bc dircctly observcd by

means of the technique of electron focusing, pionccred

in mctals by Sharvin

1

and Tsoi

21

. In metals, electron

Superlattices and Microstructures, l/o/. 5, Λ/ο. 1, 1989

(a) B>Bont 131 -0.4 -0.2 0.0 0.2 wave number (1/nm) 0.4 Bt h r e s<B<Bc r l t -0.1 0.0 0.1 wave number (1/nm) 02 (c) B<B_Ihresthr, -0.2 -0.1 0.0 0.1 wave number (1/nm)

Fäg.4. Wave number dcpcndcncc of t he cncrgy

B„(/C) in the thrce field rcgimcs discusscd in thc text.

(a) W ' = 2 0 0 n m , B= l.5 T; (h) W = I O O n m ,

B= l T; (c) H ' = 7 5 n m , Β=-0.05Ύ. Edge slatcs

arc suppresscd undci thc cotulitions of Fig. 4c. The

horizontal line at !6.9mcV indicatcs thc Fcrmi

en-crgy. (These numcrical valucs conespond to the

cxpcriments of Ref. 6 ) Thc solid pnits of the cui vcs

a i c cnlculatcd frei m Ihc Bnlir-Somincrfeld

quantization rulc, Ihc dashcd p a i l s are

intcrpo-lations. Thc shadcd arca is thc rcgion of classical

skipping orbils, and is houndcd hy (hc l wo

parabolas shown in Fig. 3 (wilh Ihc cotrcspondence

k = - XetilK).

injcctor to collcctor consists of skipping orbits (for a

spccularly rcflecting boundary). Focusing occurs if thc

point contact Separation L is an integer multiple of

thc cyclotron diamctcr, that is for fickls B which arc

multiples of

hk

F

The classical focusing spcctrum consists of a scrics of

pcaks in the collcclor voltagc of equal hcight and

constant Separation 5

rocus

. Such a spcctrum is

com-monly obscivcd in mctals

22

, albcit wilh a dccrcasing

hcight of subscqucnt peaks bccause of partially diffuse

scattering.

Thc clcctron focusing spcctrum in a 2DEG,

rc-portcd in Ref. 7, is strikingly diffcrcnt. At low fickls

a series of focusing pcaks is, indccd, obscrvcd at the

cxpcctcd positions — dcmonstrating spccular

re-flcction at thc mirroi foimcd by thc gatc polcntial.

Howevci, fincslruclurc is supciimposcd on thc

focus-ing pcaks at low tcmpcialurcs. Morcovcr, at highci

fickls Ihc collccloi voltagc shows oscillalions with a

much larger amplitude than thc low-ficld focusing

pcaks, although retaining Bfocm äs thc dominant

pcriodicity. As we havc shown in Ref. 8, thc

intcrfer-ence at thc collcctor of diffcrcnt phasc cohercnt

skip-ping orbits can explain the csscntial fcatures of the

cxpcriments. The diffcrcnce bctwccn cohercnt and

classical electron focusing is onc of length scalcs: The

ratios l

r

\W and λ

} \L arc, rcspcctivcly, l O2

and l O

4

timcs larger in thc 2DEG than in a typical metal! The

significance of CEF is that it dcmonstratcs that an

interfcrencc expeiimcnt can bc rcalizcd with a QPC

äs point source and dctcctor.

In this scction we would likc to discuss CEF from

a diffcrent point of vicw, äs a typical example of a

non-local voltagc measurcment

23

. The injector is thc

current source, and the collcctor a voltagc probe. The

non-locality of thc voltagc mcasurcmcnt manifests

it-sclf in thc depcndcnce of thc collcctor voltagc V

c

on

132

Superlattices and Microstructures, Vol. 5, Λ/ο. 7, 1989

-0.5

0.3

Fig.5. Electron focusing spcclrum showing

focus-ing pcak.s at multiples of fi

rocll

, ss 0.066 T (Eq. (5)

with k

F

= l . 5 x lO^m"' and /,= 3.0 μττι). Note thc

intcrfcrcncc fringes. For rcvcisc ficlds Ihe normal

Hall rcsistance is sccn. Thc insct givcs thc

cxpcr-imcntal configuralion, with thc gatc dcfining thc

injcctor and collector point contacts and thc 2DEG

boundary shown in hlack. Thc two cxpcrimcntal

traccs corrcspond to intcrchangcd currcnt and

voltagc Icads, and dcmonsttatc thc

injcctor-collcc-tor reciprocity. (Fig. takcn from Ref. 23.)

Fig. 5 (with /,· Ihc injectcd currcnt) is altcrnatingly

smaller and largcr than its normal valuc (Eq. (3))

which is obscrvcd in rcvcrsc ficlds. [Note the fine

structurc on the focusing pcaks; Thc largo high-ficld

oscillations mcntioncd above are outside thc ränge of

this figurc.] Fig. .5 contains two cxperimcntal traccs

(onc with focusing pcaks for positive B and onc for

negative B), which wcre obtaincd upon intcrchanging

currcnt and voltage leads — so that the injector

be-comcs thc collector and vice versa. Thc reciprocity of

injcctor and collector is evident and is in agrecmcnt

with the reciprocity rclation for four-tcrminal phase

cohcrcnt conductanccs which was rcccntly dcrivcd by

Büttiker

q

. This niccly dcmonstratcs thc validity of

symmctrics of thc Onsager-Casimir type in non-local

voltagc mcasurcmcnts (scc Ref. 20 for othcr

expcr-imcntal confirmations).

Acknowledgement- This articlc is bascd on work done

in collaboration with H. Ahmed, M.E.I. Brockaart,

C.T. Foxon, J..I. Harris, L.P. Kouwcnhovcn, P.H.M.

van Loosdrccht, D. van der Marel, J.E. Mooij, M.

Pcppcr, T..I. Thornton, and J.G. Williamson. We

t h a n k .I.A. Pals and M.F.H. Schuurmans for support.

References

1. Yu.V. Sharvin, Soviel Physics JET P 21, 655

(1965). For a rcvicw, sce A.G.M. Janscn, A.P. van

Gelder, and P. Wydcr, Journal of Physics C 13,

6073(1980).

2. B..I. van Wecs, H. van Houtcn, C.W..I. Bccnakkcr,

J.G. Williamson, L.P. Kouwcnhovcn, D. van der

Marcl, and C.T. Foxon, Physical Review Lettern

60, 848 (1988); B..I. van Wccs, et αι., Physieal

Re-view B (to bc publishcd).

3. D.A. Wharam, T..I. Thornton, R. Ncwbury, M.

Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko,

D.C. Pcacock, D.A. Ritchic, and G.A.C. Joncs,

Journal of Physics C 21, L209 (1988).

4. G. Timp, A.M. Chang, P. Mankicwich, R.

Bchringcr, J.E. Cunningham, T.Y. Chang, and

R.E. Howard, Physical Review Letter.·; 59, 732

(1987).

5. C.W.J. Beenakker and H. van Houten, Physical

Review Leiters 60, 2406 (1988).

6. M.L. Roukes, A. Scherer, S.J. Allen, Jr., H.G.

Craighcad, R.M. Ruthen, E.D. Bccbc, and J.P.

Harbison, Physical Review Leiters 59, 3011 (1987).

7. H. van Houtcn, B.J. van Wccs, J.E. Mooij, C.W..1.

Beenakker, J.G. Williamson, and C.T. Foxon,

Europhysics Leiters 5, 721 (1988).

8. C.W.J. Beenakker, H. van Houten, and B.J. van

Wccs, Europhysics Leiters (to be publishcd).

9. M. Büttikcr, Physical Rcview lottern 57, 1761

(1986); IBM Journal of Research and Development

32,317(1988).

10. T.P. Smith, I I I , .I.A. Brum, J.M. Hong, C.M.

Knocdlcr, H. Arnot, and L. Esaki (prcprint).

1 1 . Alternative cxplanations havc bcen givcn by D.A.

Wharam et al. (Ref. 3), by Y. Isawa (prcprint), and

by R. Johnston and L. Schweitzcr (prcprint).

17. R. Landauer, IBM Journal of Research and

De-velopment \, 223 (1957); 32, 306 (1988); Zeitschrift

für Physik 568, 217(1987).

13. Y. Imry, in Directions in Condensed Matter

Physics, Vol. l, cditcd by G. Grinstein and G.

Mazenko (World Scicntific, Singaporc, 1986) pagc

129.

14. E.G. Haanappcl and D. van der Marcl (prcprint);

A.D. Stone and A. Szafer (private communication).

15. M. Büttiker (prcprint).

16. H. van Houten, C.W.J. Beenakker, P.H.M. van

Loosdrccht, T.J. Thornton, H. Ahmed, M. Pcppcr,

C.T. Foxon, and J..I. Harris, Physical Review B 37,

8534(1988).

17. A discussion of this controvcrsy is given by A.D.

Stonc and A. Szafer, IBM Journal of Research and

Development 32, 384 (1988).

18. C.J.B. Ford, T.J. Thornton, R. Ncwbury, M.

Pcppcr, H. Ahmed, D.C. Pcacock, D.A. Ritchic,

J.E.F. Frost, and G.A.C. Joncs (prcprint).

19. F.M. Pcctcrs, Physical Review Leiters 6 ) , 589

(1988).

20. A.D. Bcnoit, S. Washburn, C.P. Umbach, R.B.

Laibowitz, and R.A. Wcbb, Physical Review Leiters

57, 176.5 (1986); G. Timp, H.U. Baranger, P.

dcVcgvar, J.E. C u n n i n g h a m , R.E. Howard, R.

Bchringcr, and P.M. Mankicwich, Physical Review

Leiters 60, 2081 (1988).

21. V.S. Tsoi, JETP Leiters 19, 70 (1974).

22. P.A.M. Benistant, G.F.A. van de Walle, H. van

Kempen, and P. Wydcr, Physical Review B 33, 690

(1986).